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So I know how to use goodness of fit tests generally. For example if I wanted to check whether my data came from a Poisson distribution I would first estimate the parameter using MLE (so just the sample average) and then apply goodness of fit tests.

But what about for a distribution like student-t, where the degree of freedom parameter is typically an integer and the MLE doesn't have a closed form solution (I think).

So I would imagine I would have,say, 1000 observations. I think my first step would be generating an estimate for the degree of freedom parameter (using MLE) and then be on my way doing the goodness of fit tests.

Please feel free to comment on this approach or what you would do differently.

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  • $\begingroup$ Theoretically you can restrict the parameter space to be positive integer, and still can maximize that, at least numerically. In practice I have not try that before, and maybe that is not numerically stable, I am not so sure. I think the general approach is theoretically sound. Besides the chi-square test, you may also look for other like K-S test and etc which are also very common. $\endgroup$
    – BGM
    Commented May 23, 2016 at 5:57
  • $\begingroup$ Would you have any idea of how I would get the MLE estimate in excel. I thought solver would work but it doesnt optimize at all. Like say column A is filled with 1000 values (ranging from -3 to 3) and in column B I have 1000 student-t probabilities base on column A values. I.e. =t-dist("A1", C1, FALSE) where C1 is another cell where I have the d.f parameter to optimize. Then C2 equals the product of those 1000 numbers in column B, effectively giving the likelihood. But solver doesnt optimize. $\endgroup$ Commented May 23, 2016 at 16:57

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